Locality Sensitive Hashing For Set-queries Motivated By Group Recommendations
Kaplan Haim, Tenenbaum Jay. Arxiv 2020
Locality Sensitive Hashing (LSH) is an effective method to index a set of points such that we can efficiently find the nearest neighbors of a query point. We extend this method to our novel Set-query LSH (SLSH), such that it can find the nearest neighbors of a set of points, given as a query. Let \( s(x,y) \) be the similarity between two points \( x \) and \( y \). We define a similarity between a set \( Q\) and a point \( x \) by aggregating the similarities \( s(p,x) \) for all \( p\in Q \). For example, we can take \( s(p,x) \) to be the angular similarity between \( p \) and \( x \) (i.e., \(1-{\angle (x,p)}/{\pi}\)), and aggregate by arithmetic or geometric averaging, or taking the lowest similarity. We develop locality sensitive hash families and data structures for a large set of such arithmetic and geometric averaging similarities, and analyze their collision probabilities. We also establish an analogous framework and hash families for distance functions. Specifically, we give a structure for the euclidean distance aggregated by either averaging or taking the maximum. We leverage SLSH to solve a geometric extension of the approximate near neighbors problem. In this version, we consider a metric for which the unit ball is an ellipsoid and its orientation is specified with the query. An important application that motivates our work is group recommendation systems. Such a system embeds movies and users in the same feature space, and the task of recommending a movie for a group to watch together, translates to a set-query \( Q \) using an appropriate similarity.
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